trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies ...

, the law of cotangentsThe Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960. is a relationship among the lengths of the sides of a triangle and the cotangents of the halves of the three angles. This is also known as the Cot Theorem. Just as three quantities whose equality is expressed by the

law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and ar ...

are equal to the diameter of the

circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

of the triangle (or to its reciprocal, depending on how the law is expressed), so also the law of cotangents relates the radius of the inscribed circle of a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

(the

inradius In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

) to its sides and angles.

Statement

Using the usual notations for a triangle (see the figure at the upper right), where , , are the lengths of the three sides, , , are the vertices opposite those three respective sides, , , are the corresponding angles at those vertices, is the semi-perimeter, that is, , and is the radius of the inscribed circle, the law of cotangents states that :

\frac = \frac = \frac = \frac\,

and furthermore that the inradius is given by :

r = \sqrt\,.

Proof

In the upper figure, the points of tangency of the incircle with the sides of the triangle break the perimeter into 6 segments, in 3 pairs. In each pair the segments are of equal length. For example, the 2 segments adjacent to vertex are equal. If we pick one segment from each pair, their sum will be the semiperimeter . An example of this is the segments shown in color in the figure. The two segments making up the red line add up to , so the blue segment must be of length . Obviously, the other five segments must also have lengths , , or , as shown in the lower figure. By inspection of the figure, using the definition of the cotangent function, we have :

\cot\left(\frac\right) =\frac\,

and similarly for the other two angles, proving the first assertion. For the second one—the inradius formula—we start from the general addition formula: :

\cot (u+v+w) = \frac.

Applying to , we obtain: :

\cot\left(\frac\right)  \cot \left(\frac\right)  \cot \left(\frac\right) = \cot\left(\frac\right) + \cot \left(\frac\right) + \cot \left(\frac\right).

(This is also the triple cotangent identity) Substituting the values obtained in the first part, we get: :

\frac r \frac r \frac r = \frac r + \frac r +\frac r =\fracr=\fracr.

Multiplying through by gives the value of , proving the second assertion.

Some proofs using the law of cotangents

A number of other results can be derived from the law of cotangents. *

Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...

. Note that the area of triangle is also divided into 6 smaller triangles, also in 3 pairs, with the triangles in each pair having the same area. For example, the two triangles near vertex , being right triangles of width and height , each have an area of . So those two triangles together have an area of , and the area of the whole triangle is therefore

S = r(s-a) + r(s-b) + r(s-c) = r\bigl(3s - (a+b+c)\bigr) = r(3s - 2s) = rs

This gives the result

S = \sqrt

as required. * Mollweide's first formula. From the addition formula and the law of cotangents we have

\frac  = \frac =  \frac .

This gives the result

\dfrac =\dfrac

as required. * Mollweide's second formula. From the addition formula and the law of cotangents we have

= & \frac = \frac . \end

Here, an extra step is required to transform a product into a sum, according to the sum/product formula. This gives the result

\dfrac  = \dfrac

as required. *The

law of tangents In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, , , and are the lengths of the three sides of the triangle, and , , ...

can also be derived from this .

References

* {{refend Trigonometry Theorems about triangles

Statement

Proof

Some proofs using the law of cotangents

See also

References